SECTION 11.4 Matrix Algebra 795 11.4 Matrix Algebra Section 11.2 defined a matrix as a rectangular array of real numbers and used an augmented matrix to represent a system of linear equations. There is, however, a branch of mathematics, called linear algebra , in which an algebra of matrices is defined. This section is a survey of how this matrix algebra is developed. Before getting started, recall the definition of a matrix. OBJECTIVES 1 Find the Sum and Difference of Two Matrices (p. 796) 2 Find Scalar Multiples of a Matrix (p. 798) 3 Find the Product of Two Matrices (p. 799) 4 Find the Inverse of a Matrix (p. 803) 5 Solve a System of Linear Equations Using an Inverse Matrix (p. 807) DEFINITION Matrix A matrix is defined as a rectangular array of numbers: j n m Column 1 Column 2 Column Column Row 1 Row 2 Row Row ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ a a a a a a a a a a a a a a a a i j n j n i i ij in m m mj mn 11 12 1 1 21 22 2 2 1 2 1 2 In Words In a matrix, the rows go across, and the columns go down. Each number aij of the matrix has two indexes: the row index i and the column index j. The matrix shown here has m rows and n columns.The numbers aij are usually referred to as the entries of the matrix. For example, a23 refers to the entry in the second row, third column. Arranging Data in a Matrix In a survey of 900 people, the following information was obtained: 200 men Thought federal defense spending was too high 150 men Thought federal defense spending was too low 45 men Had no opinion 315 women Thought federal defense spending was too high 125 women Thought federal defense spending was too low 65 women Had no opinion EXAMPLE 1 (continued) 75. The function ( ) ( ) = + − f x x 3 log 1 5 is one-to-one. Find −f .1 76. Find the distance between the vertices of ( )= − + f x x x 2 12 20 2 and ( )=− − − g x x x 3 30 77. 2 77. Expand: ( ) −x2 5 3 78. Rationalize the numerator: + − x x 7 10 79. Find an equation of the line perpendicular to ( ) = − + f x x 2 5 7 where = x 10.
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